Normal states of type III factors

Yasuyuki Kawahigashi, Yoshiko Ogata and Erling Størmer

Vol. 267 (2014), No. 1, 131–139

DOI: 10.2140/pjm.2014.267.131

Abstract

Let $M$ be a factor of type III with separable predual and with normal states ${\phi }_1,\dots ,{\phi }_k,\omega$ with $\omega$ faithful. Let $A$ be a finite-dimensional $C^{\ast }$-subalgebra of $M$. Then it is shown that there is a unitary operator $u\in M$ such that ${\phi }_i\circ Adu=\omega$ on $A$ for $i=1,\dots ,k$. This follows from an embedding result of a finite-dimensional $C^{\ast }$-algebra with a faithful state into $M$ with finitely many given states. We also give similar embedding results of $C^{\ast }$-algebras and von Neumann algebras with faithful states into $M$. Another similar result for a factor of type II${}_1$ instead of type III holds.

Dedicated to Masamichi Takesaki on the occasion of his eightieth birthday.

Keywords

von Neumann algebra, type III factor, normal state

Mathematical Subject Classification 2010

Primary: 46L30

Milestones

Received: 27 February 2013

Revised: 30 May 2013

Accepted: 7 June 2013

Published: 22 December 2013